LMFDB - Embedded newform 4024.2.a.g.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.1318017734$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.741660 q^{3} -2.54437 q^{5} -0.0722362 q^{7} -2.44994 q^{9} +O(q^{10})$	$q-0.741660 q^{3} -2.54437 q^{5} -0.0722362 q^{7} -2.44994 q^{9} -3.91101 q^{11} -1.31968 q^{13} +1.88705 q^{15} +4.63073 q^{17} +3.41710 q^{19} +0.0535746 q^{21} -3.73992 q^{23} +1.47380 q^{25} +4.04200 q^{27} -6.95019 q^{29} -8.64883 q^{31} +2.90064 q^{33} +0.183795 q^{35} -6.78962 q^{37} +0.978752 q^{39} +2.07592 q^{41} -4.11353 q^{43} +6.23355 q^{45} -0.0199974 q^{47} -6.99478 q^{49} -3.43442 q^{51} -5.41774 q^{53} +9.95104 q^{55} -2.53432 q^{57} +2.00640 q^{59} +3.92630 q^{61} +0.176974 q^{63} +3.35775 q^{65} -11.5835 q^{67} +2.77374 q^{69} +9.21429 q^{71} +5.01924 q^{73} -1.09306 q^{75} +0.282516 q^{77} -6.85436 q^{79} +4.35203 q^{81} +9.70331 q^{83} -11.7823 q^{85} +5.15467 q^{87} +6.22827 q^{89} +0.0953285 q^{91} +6.41449 q^{93} -8.69435 q^{95} +12.4866 q^{97} +9.58174 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.741660	−0.428197	−0.214099	−	0.976812i	$-0.568681\pi$
$3$	−0.741660	−0.428197	−0.214099	+	0.976812i	$0.568681\pi$
$4$	0	0
$5$	−2.54437	−1.13788	−0.568938	−	0.822381i	$-0.692645\pi$
$5$	−2.54437	−1.13788	−0.568938	+	0.822381i	$0.692645\pi$
$6$	0	0
$7$	−0.0722362	−0.0273027	−0.0136514	−	0.999907i	$-0.504345\pi$
$7$	−0.0722362	−0.0273027	−0.0136514	+	0.999907i	$0.504345\pi$
$8$	0	0
$9$	−2.44994	−0.816647
$10$	0	0
$11$	−3.91101	−1.17921	−0.589607	−	0.807690i	$-0.700717\pi$
$11$	−3.91101	−1.17921	−0.589607	+	0.807690i	$0.700717\pi$
$12$	0	0
$13$	−1.31968	−0.366013	−0.183006	−	0.983112i	$-0.558583\pi$
$13$	−1.31968	−0.366013	−0.183006	+	0.983112i	$0.558583\pi$
$14$	0	0
$15$	1.88705	0.487235
$16$	0	0
$17$	4.63073	1.12312	0.561558	−	0.827437i	$-0.310202\pi$
$17$	4.63073	1.12312	0.561558	+	0.827437i	$0.310202\pi$
$18$	0	0
$19$	3.41710	0.783936	0.391968	−	0.919979i	$-0.371794\pi$
$19$	3.41710	0.783936	0.391968	+	0.919979i	$0.371794\pi$
$20$	0	0
$21$	0.0535746	0.0116909
$22$	0	0
$23$	−3.73992	−0.779826	−0.389913	−	0.920852i	$-0.627495\pi$
$23$	−3.73992	−0.779826	−0.389913	+	0.920852i	$0.627495\pi$
$24$	0	0
$25$	1.47380	0.294760
$26$	0	0
$27$	4.04200	0.777883
$28$	0	0
$29$	−6.95019	−1.29062	−0.645309	−	0.763922i	$-0.723271\pi$
$29$	−6.95019	−1.29062	−0.645309	+	0.763922i	$0.723271\pi$
$30$	0	0
$31$	−8.64883	−1.55338	−0.776688	−	0.629886i	$-0.783102\pi$
$31$	−8.64883	−1.55338	−0.776688	+	0.629886i	$0.783102\pi$
$32$	0	0
$33$	2.90064	0.504936
$34$	0	0
$35$	0.183795	0.0310671
$36$	0	0
$37$	−6.78962	−1.11621	−0.558104	−	0.829771i	$-0.688471\pi$
$37$	−6.78962	−1.11621	−0.558104	+	0.829771i	$0.688471\pi$
$38$	0	0
$39$	0.978752	0.156726
$40$	0	0
$41$	2.07592	0.324204	0.162102	−	0.986774i	$-0.448173\pi$
$41$	2.07592	0.324204	0.162102	+	0.986774i	$0.448173\pi$
$42$	0	0
$43$	−4.11353	−0.627308	−0.313654	−	0.949537i	$-0.601553\pi$
$43$	−4.11353	−0.627308	−0.313654	+	0.949537i	$0.601553\pi$
$44$	0	0
$45$	6.23355	0.929243
$46$	0	0
$47$	−0.0199974	−0.00291692	−0.00145846	−	0.999999i	$-0.500464\pi$
$47$	−0.0199974	−0.00291692	−0.00145846	+	0.999999i	$0.500464\pi$
$48$	0	0
$49$	−6.99478	−0.999255
$50$	0	0
$51$	−3.43442	−0.480916
$52$	0	0
$53$	−5.41774	−0.744184	−0.372092	−	0.928196i	$-0.621359\pi$
$53$	−5.41774	−0.744184	−0.372092	+	0.928196i	$0.621359\pi$
$54$	0	0
$55$	9.95104	1.34180
$56$	0	0
$57$	−2.53432	−0.335679
$58$	0	0
$59$	2.00640	0.261211	0.130605	−	0.991434i	$-0.458308\pi$
$59$	2.00640	0.261211	0.130605	+	0.991434i	$0.458308\pi$
$60$	0	0
$61$	3.92630	0.502711	0.251355	−	0.967895i	$-0.419124\pi$
$61$	3.92630	0.502711	0.251355	+	0.967895i	$0.419124\pi$
$62$	0	0
$63$	0.176974	0.0222967
$64$	0	0
$65$	3.35775	0.416477
$66$	0	0
$67$	−11.5835	−1.41515	−0.707575	−	0.706638i	$-0.750211\pi$
$67$	−11.5835	−1.41515	−0.707575	+	0.706638i	$0.750211\pi$
$68$	0	0
$69$	2.77374	0.333920
$70$	0	0
$71$	9.21429	1.09354	0.546768	−	0.837284i	$-0.315858\pi$
$71$	9.21429	1.09354	0.546768	+	0.837284i	$0.315858\pi$
$72$	0	0
$73$	5.01924	0.587458	0.293729	−	0.955889i	$-0.405104\pi$
$73$	5.01924	0.587458	0.293729	+	0.955889i	$0.405104\pi$
$74$	0	0
$75$	−1.09306	−0.126216
$76$	0	0
$77$	0.282516	0.0321957
$78$	0	0
$79$	−6.85436	−0.771175	−0.385588	−	0.922671i	$-0.626001\pi$
$79$	−6.85436	−0.771175	−0.385588	+	0.922671i	$0.626001\pi$
$80$	0	0
$81$	4.35203	0.483559
$82$	0	0
$83$	9.70331	1.06508	0.532538	−	0.846406i	$-0.321238\pi$
$83$	9.70331	1.06508	0.532538	+	0.846406i	$0.321238\pi$
$84$	0	0
$85$	−11.7823	−1.27797
$86$	0	0
$87$	5.15467	0.552639
$88$	0	0
$89$	6.22827	0.660196	0.330098	−	0.943947i	$-0.392918\pi$
$89$	6.22827	0.660196	0.330098	+	0.943947i	$0.392918\pi$
$90$	0	0
$91$	0.0953285	0.00999314
$92$	0	0
$93$	6.41449	0.665151
$94$	0	0
$95$	−8.69435	−0.892021
$96$	0	0
$97$	12.4866	1.26782	0.633910	−	0.773407i	$-0.281449\pi$
$97$	12.4866	1.26782	0.633910	+	0.773407i	$0.281449\pi$
$98$	0	0
$99$	9.58174	0.963001
$100$	0	0
$101$	4.40520	0.438334	0.219167	−	0.975687i	$-0.429666\pi$
$101$	4.40520	0.438334	0.219167	+	0.975687i	$0.429666\pi$
$102$	0	0
$103$	14.4328	1.42211	0.711053	−	0.703138i	$-0.248218\pi$
$103$	14.4328	1.42211	0.711053	+	0.703138i	$0.248218\pi$
$104$	0	0
$105$	−0.136314	−0.0133028
$106$	0	0
$107$	13.3008	1.28583	0.642917	−	0.765936i	$-0.277724\pi$
$107$	13.3008	1.28583	0.642917	+	0.765936i	$0.277724\pi$
$108$	0	0
$109$	−4.99922	−0.478839	−0.239419	−	0.970916i	$-0.576957\pi$
$109$	−4.99922	−0.478839	−0.239419	+	0.970916i	$0.576957\pi$
$110$	0	0
$111$	5.03559	0.477957
$112$	0	0
$113$	8.18625	0.770098	0.385049	−	0.922896i	$-0.374185\pi$
$113$	8.18625	0.770098	0.385049	+	0.922896i	$0.374185\pi$
$114$	0	0
$115$	9.51572	0.887345
$116$	0	0
$117$	3.23313	0.298903
$118$	0	0
$119$	−0.334506	−0.0306641
$120$	0	0
$121$	4.29600	0.390545
$122$	0	0
$123$	−1.53962	−0.138823
$124$	0	0
$125$	8.97194	0.802475
$126$	0	0
$127$	0.475317	0.0421775	0.0210888	−	0.999778i	$-0.493287\pi$
$127$	0.475317	0.0421775	0.0210888	+	0.999778i	$0.493287\pi$
$128$	0	0
$129$	3.05084	0.268612
$130$	0	0
$131$	4.04171	0.353125	0.176563	−	0.984289i	$-0.443502\pi$
$131$	4.04171	0.353125	0.176563	+	0.984289i	$0.443502\pi$
$132$	0	0
$133$	−0.246838	−0.0214036
$134$	0	0
$135$	−10.2843	−0.885135
$136$	0	0
$137$	−7.70993	−0.658704	−0.329352	−	0.944207i	$-0.606830\pi$
$137$	−7.70993	−0.658704	−0.329352	+	0.944207i	$0.606830\pi$
$138$	0	0
$139$	9.65174	0.818650	0.409325	−	0.912389i	$-0.365764\pi$
$139$	9.65174	0.818650	0.409325	+	0.912389i	$0.365764\pi$
$140$	0	0
$141$	0.0148313	0.00124902
$142$	0	0
$143$	5.16127	0.431607
$144$	0	0
$145$	17.6838	1.46856
$146$	0	0
$147$	5.18775	0.427878
$148$	0	0
$149$	−3.35133	−0.274552	−0.137276	−	0.990533i	$-0.543835\pi$
$149$	−3.35133	−0.274552	−0.137276	+	0.990533i	$0.543835\pi$
$150$	0	0
$151$	−0.0736439	−0.00599305	−0.00299653	−	0.999996i	$-0.500954\pi$
$151$	−0.0736439	−0.00599305	−0.00299653	+	0.999996i	$0.500954\pi$
$152$	0	0
$153$	−11.3450	−0.917190
$154$	0	0
$155$	22.0058	1.76755
$156$	0	0
$157$	−13.6271	−1.08756	−0.543779	−	0.839229i	$-0.683007\pi$
$157$	−13.6271	−1.08756	−0.543779	+	0.839229i	$0.683007\pi$
$158$	0	0
$159$	4.01812	0.318658
$160$	0	0
$161$	0.270157	0.0212914
$162$	0	0
$163$	−7.76666	−0.608331	−0.304166	−	0.952619i	$-0.598378\pi$
$163$	−7.76666	−0.608331	−0.304166	+	0.952619i	$0.598378\pi$
$164$	0	0
$165$	−7.38029	−0.574555
$166$	0	0
$167$	15.2145	1.17733	0.588665	−	0.808377i	$-0.299654\pi$
$167$	15.2145	1.17733	0.588665	+	0.808377i	$0.299654\pi$
$168$	0	0
$169$	−11.2584	−0.866035
$170$	0	0
$171$	−8.37168	−0.640199
$172$	0	0
$173$	13.5942	1.03355	0.516775	−	0.856121i	$-0.327132\pi$
$173$	13.5942	1.03355	0.516775	+	0.856121i	$0.327132\pi$
$174$	0	0
$175$	−0.106462	−0.00804776
$176$	0	0
$177$	−1.48806	−0.111850
$178$	0	0
$179$	0.670414	0.0501091	0.0250545	−	0.999686i	$-0.492024\pi$
$179$	0.670414	0.0501091	0.0250545	+	0.999686i	$0.492024\pi$
$180$	0	0
$181$	−2.71566	−0.201854	−0.100927	−	0.994894i	$-0.532181\pi$
$181$	−2.71566	−0.201854	−0.100927	+	0.994894i	$0.532181\pi$
$182$	0	0
$183$	−2.91198	−0.215260
$184$	0	0
$185$	17.2753	1.27010
$186$	0	0
$187$	−18.1108	−1.32439
$188$	0	0
$189$	−0.291979	−0.0212383
$190$	0	0
$191$	17.6633	1.27807	0.639034	−	0.769178i	$-0.279334\pi$
$191$	17.6633	1.27807	0.639034	+	0.769178i	$0.279334\pi$
$192$	0	0
$193$	−3.48681	−0.250986	−0.125493	−	0.992094i	$-0.540051\pi$
$193$	−3.48681	−0.250986	−0.125493	+	0.992094i	$0.540051\pi$
$194$	0	0
$195$	−2.49030	−0.178334
$196$	0	0
$197$	−7.46549	−0.531894	−0.265947	−	0.963988i	$-0.585685\pi$
$197$	−7.46549	−0.531894	−0.265947	+	0.963988i	$0.585685\pi$
$198$	0	0
$199$	7.26038	0.514674	0.257337	−	0.966322i	$-0.417155\pi$
$199$	7.26038	0.514674	0.257337	+	0.966322i	$0.417155\pi$
$200$	0	0
$201$	8.59102	0.605964
$202$	0	0
$203$	0.502055	0.0352373
$204$	0	0
$205$	−5.28190	−0.368904
$206$	0	0
$207$	9.16257	0.636843
$208$	0	0
$209$	−13.3643	−0.924428
$210$	0	0
$211$	−16.3882	−1.12821	−0.564107	−	0.825702i	$-0.690779\pi$
$211$	−16.3882	−1.12821	−0.564107	+	0.825702i	$0.690779\pi$
$212$	0	0
$213$	−6.83387	−0.468249
$214$	0	0
$215$	10.4663	0.713798
$216$	0	0
$217$	0.624758	0.0424114
$218$	0	0
$219$	−3.72257	−0.251548
$220$	0	0
$221$	−6.11107	−0.411075
$222$	0	0
$223$	19.7349	1.32155	0.660773	−	0.750586i	$-0.270229\pi$
$223$	19.7349	1.32155	0.660773	+	0.750586i	$0.270229\pi$
$224$	0	0
$225$	−3.61073	−0.240715
$226$	0	0
$227$	14.6923	0.975160	0.487580	−	0.873078i	$-0.337880\pi$
$227$	14.6923	0.975160	0.487580	+	0.873078i	$0.337880\pi$
$228$	0	0
$229$	−20.3696	−1.34606	−0.673030	−	0.739615i	$-0.735007\pi$
$229$	−20.3696	−1.34606	−0.673030	+	0.739615i	$0.735007\pi$
$230$	0	0
$231$	−0.209531	−0.0137861
$232$	0	0
$233$	6.82915	0.447392	0.223696	−	0.974659i	$-0.428188\pi$
$233$	6.82915	0.447392	0.223696	+	0.974659i	$0.428188\pi$
$234$	0	0
$235$	0.0508807	0.00331909
$236$	0	0
$237$	5.08360	0.330215
$238$	0	0
$239$	−11.7576	−0.760533	−0.380266	−	0.924877i	$-0.624168\pi$
$239$	−11.7576	−0.760533	−0.380266	+	0.924877i	$0.624168\pi$
$240$	0	0
$241$	−22.0162	−1.41819	−0.709095	−	0.705113i	$-0.750896\pi$
$241$	−22.0162	−1.41819	−0.709095	+	0.705113i	$0.750896\pi$
$242$	0	0
$243$	−15.3537	−0.984942
$244$	0	0
$245$	17.7973	1.13703
$246$	0	0
$247$	−4.50947	−0.286931
$248$	0	0
$249$	−7.19655	−0.456063
$250$	0	0
$251$	−18.3949	−1.16108	−0.580538	−	0.814233i	$-0.697158\pi$
$251$	−18.3949	−1.16108	−0.580538	+	0.814233i	$0.697158\pi$
$252$	0	0
$253$	14.6268	0.919582
$254$	0	0
$255$	8.73844	0.547222
$256$	0	0
$257$	−10.7431	−0.670138	−0.335069	−	0.942194i	$-0.608760\pi$
$257$	−10.7431	−0.670138	−0.335069	+	0.942194i	$0.608760\pi$
$258$	0	0
$259$	0.490456	0.0304755
$260$	0	0
$261$	17.0275	1.05398
$262$	0	0
$263$	22.6383	1.39593	0.697967	−	0.716130i	$-0.254088\pi$
$263$	22.6383	1.39593	0.697967	+	0.716130i	$0.254088\pi$
$264$	0	0
$265$	13.7847	0.846789
$266$	0	0
$267$	−4.61926	−0.282694
$268$	0	0
$269$	−26.5000	−1.61573	−0.807865	−	0.589367i	$-0.799377\pi$
$269$	−26.5000	−1.61573	−0.807865	+	0.589367i	$0.799377\pi$
$270$	0	0
$271$	25.5814	1.55396	0.776979	−	0.629527i	$-0.216751\pi$
$271$	25.5814	1.55396	0.776979	+	0.629527i	$0.216751\pi$
$272$	0	0
$273$	−0.0707013	−0.00427904
$274$	0	0
$275$	−5.76405	−0.347586
$276$	0	0
$277$	23.1242	1.38940	0.694699	−	0.719301i	$-0.255538\pi$
$277$	23.1242	1.38940	0.694699	+	0.719301i	$0.255538\pi$
$278$	0	0
$279$	21.1891	1.26856
$280$	0	0
$281$	23.0304	1.37388	0.686938	−	0.726716i	$-0.258954\pi$
$281$	23.0304	1.37388	0.686938	+	0.726716i	$0.258954\pi$
$282$	0	0
$283$	20.1137	1.19563	0.597817	−	0.801633i	$-0.296035\pi$
$283$	20.1137	1.19563	0.597817	+	0.801633i	$0.296035\pi$
$284$	0	0
$285$	6.44825	0.381961
$286$	0	0
$287$	−0.149956	−0.00885165
$288$	0	0
$289$	4.44365	0.261391
$290$	0	0
$291$	−9.26078	−0.542877
$292$	0	0
$293$	−23.3320	−1.36307	−0.681536	−	0.731785i	$-0.738688\pi$
$293$	−23.3320	−1.36307	−0.681536	+	0.731785i	$0.738688\pi$
$294$	0	0
$295$	−5.10501	−0.297225
$296$	0	0
$297$	−15.8083	−0.917291
$298$	0	0
$299$	4.93549	0.285426
$300$	0	0
$301$	0.297146	0.0171272
$302$	0	0
$303$	−3.26716	−0.187693
$304$	0	0
$305$	−9.98994	−0.572022
$306$	0	0
$307$	24.4643	1.39625	0.698125	−	0.715976i	$-0.254018\pi$
$307$	24.4643	1.39625	0.698125	+	0.715976i	$0.254018\pi$
$308$	0	0
$309$	−10.7042	−0.608942
$310$	0	0
$311$	27.6677	1.56889	0.784446	−	0.620197i	$-0.212947\pi$
$311$	27.6677	1.56889	0.784446	+	0.620197i	$0.212947\pi$
$312$	0	0
$313$	14.6198	0.826359	0.413179	−	0.910650i	$-0.364418\pi$
$313$	14.6198	0.826359	0.413179	+	0.910650i	$0.364418\pi$
$314$	0	0
$315$	−0.450288	−0.0253708
$316$	0	0
$317$	−25.6230	−1.43913	−0.719567	−	0.694423i	$-0.755660\pi$
$317$	−25.6230	−1.43913	−0.719567	+	0.694423i	$0.755660\pi$
$318$	0	0
$319$	27.1822	1.52191
$320$	0	0
$321$	−9.86464	−0.550590
$322$	0	0
$323$	15.8236	0.880451
$324$	0	0
$325$	−1.94494	−0.107886
$326$	0	0
$327$	3.70772	0.205038
$328$	0	0
$329$	0.00144454	7.96398e−5 0
$330$	0	0
$331$	10.8790	0.597966	0.298983	−	0.954258i	$-0.403353\pi$
$331$	10.8790	0.597966	0.298983	+	0.954258i	$0.403353\pi$
$332$	0	0
$333$	16.6342	0.911547
$334$	0	0
$335$	29.4727	1.61027
$336$	0	0
$337$	−32.8691	−1.79049	−0.895247	−	0.445570i	$-0.853001\pi$
$337$	−32.8691	−1.79049	−0.895247	+	0.445570i	$0.853001\pi$
$338$	0	0
$339$	−6.07141	−0.329754
$340$	0	0
$341$	33.8257	1.83176
$342$	0	0
$343$	1.01093	0.0545851
$344$	0	0
$345$	−7.05742	−0.379959
$346$	0	0
$347$	−8.37603	−0.449649	−0.224824	−	0.974399i	$-0.572181\pi$
$347$	−8.37603	−0.449649	−0.224824	+	0.974399i	$0.572181\pi$
$348$	0	0
$349$	−20.1485	−1.07852	−0.539261	−	0.842139i	$-0.681296\pi$
$349$	−20.1485	−1.07852	−0.539261	+	0.842139i	$0.681296\pi$
$350$	0	0
$351$	−5.33414	−0.284715
$352$	0	0
$353$	28.7204	1.52863	0.764317	−	0.644841i	$-0.223076\pi$
$353$	28.7204	1.52863	0.764317	+	0.644841i	$0.223076\pi$
$354$	0	0
$355$	−23.4445	−1.24431
$356$	0	0
$357$	0.248090	0.0131303
$358$	0	0
$359$	7.51562	0.396659	0.198330	−	0.980135i	$-0.436448\pi$
$359$	7.51562	0.396659	0.198330	+	0.980135i	$0.436448\pi$
$360$	0	0
$361$	−7.32345	−0.385445
$362$	0	0
$363$	−3.18617	−0.167230
$364$	0	0
$365$	−12.7708	−0.668454
$366$	0	0
$367$	−35.1642	−1.83556	−0.917778	−	0.397094i	$-0.870019\pi$
$367$	−35.1642	−1.83556	−0.917778	+	0.397094i	$0.870019\pi$
$368$	0	0
$369$	−5.08588	−0.264760
$370$	0	0
$371$	0.391357	0.0203182
$372$	0	0
$373$	22.6922	1.17496	0.587480	−	0.809239i	$-0.300120\pi$
$373$	22.6922	1.17496	0.587480	+	0.809239i	$0.300120\pi$
$374$	0	0
$375$	−6.65413	−0.343618
$376$	0	0
$377$	9.17201	0.472383
$378$	0	0
$379$	21.3334	1.09582	0.547911	−	0.836537i	$-0.315423\pi$
$379$	21.3334	1.09582	0.547911	+	0.836537i	$0.315423\pi$
$380$	0	0
$381$	−0.352523	−0.0180603
$382$	0	0
$383$	−3.04636	−0.155662	−0.0778308	−	0.996967i	$-0.524799\pi$
$383$	−3.04636	−0.155662	−0.0778308	+	0.996967i	$0.524799\pi$
$384$	0	0
$385$	−0.718825	−0.0366347
$386$	0	0
$387$	10.0779	0.512289
$388$	0	0
$389$	−10.1215	−0.513179	−0.256589	−	0.966520i	$-0.582599\pi$
$389$	−10.1215	−0.513179	−0.256589	+	0.966520i	$0.582599\pi$
$390$	0	0
$391$	−17.3185	−0.875836
$392$	0	0
$393$	−2.99757	−0.151207
$394$	0	0
$395$	17.4400	0.877501
$396$	0	0
$397$	20.2587	1.01675	0.508377	−	0.861135i	$-0.330246\pi$
$397$	20.2587	1.01675	0.508377	+	0.861135i	$0.330246\pi$
$398$	0	0
$399$	0.183070	0.00916495
$400$	0	0
$401$	29.2699	1.46167	0.730835	−	0.682554i	$-0.239131\pi$
$401$	29.2699	1.46167	0.730835	+	0.682554i	$0.239131\pi$
$402$	0	0
$403$	11.4137	0.568556
$404$	0	0
$405$	−11.0732	−0.550230
$406$	0	0
$407$	26.5543	1.31625
$408$	0	0
$409$	−15.9671	−0.789525	−0.394762	−	0.918783i	$-0.629173\pi$
$409$	−15.9671	−0.789525	−0.394762	+	0.918783i	$0.629173\pi$
$410$	0	0
$411$	5.71815	0.282055
$412$	0	0
$413$	−0.144934	−0.00713176
$414$	0	0
$415$	−24.6888	−1.21192
$416$	0	0
$417$	−7.15830	−0.350544
$418$	0	0
$419$	−10.4908	−0.512512	−0.256256	−	0.966609i	$-0.582489\pi$
$419$	−10.4908	−0.512512	−0.256256	+	0.966609i	$0.582489\pi$
$420$	0	0
$421$	−17.1751	−0.837063	−0.418532	−	0.908202i	$-0.637455\pi$
$421$	−17.1751	−0.837063	−0.418532	+	0.908202i	$0.637455\pi$
$422$	0	0
$423$	0.0489925	0.00238210
$424$	0	0
$425$	6.82478	0.331050
$426$	0	0
$427$	−0.283621	−0.0137254
$428$	0	0
$429$	−3.82791	−0.184813
$430$	0	0
$431$	−16.8463	−0.811459	−0.405730	−	0.913993i	$-0.632983\pi$
$431$	−16.8463	−0.811459	−0.405730	+	0.913993i	$0.632983\pi$
$432$	0	0
$433$	−37.1081	−1.78330	−0.891650	−	0.452726i	$-0.850452\pi$
$433$	−37.1081	−1.78330	−0.891650	+	0.452726i	$0.850452\pi$
$434$	0	0
$435$	−13.1154	−0.628834
$436$	0	0
$437$	−12.7797	−0.611334
$438$	0	0
$439$	−1.87958	−0.0897075	−0.0448537	−	0.998994i	$-0.514282\pi$
$439$	−1.87958	−0.0897075	−0.0448537	+	0.998994i	$0.514282\pi$
$440$	0	0
$441$	17.1368	0.816038
$442$	0	0
$443$	1.51397	0.0719309	0.0359654	−	0.999353i	$-0.488549\pi$
$443$	1.51397	0.0719309	0.0359654	+	0.999353i	$0.488549\pi$
$444$	0	0
$445$	−15.8470	−0.751220
$446$	0	0
$447$	2.48555	0.117562
$448$	0	0
$449$	−1.23607	−0.0583340	−0.0291670	−	0.999575i	$-0.509285\pi$
$449$	−1.23607	−0.0583340	−0.0291670	+	0.999575i	$0.509285\pi$
$450$	0	0
$451$	−8.11894	−0.382306
$452$	0	0
$453$	0.0546187	0.00256621
$454$	0	0
$455$	−0.242551	−0.0113710
$456$	0	0
$457$	34.5452	1.61596	0.807978	−	0.589213i	$-0.200562\pi$
$457$	34.5452	1.61596	0.807978	+	0.589213i	$0.200562\pi$
$458$	0	0
$459$	18.7174	0.873654
$460$	0	0
$461$	−25.2654	−1.17673	−0.588364	−	0.808596i	$-0.700228\pi$
$461$	−25.2654	−1.17673	−0.588364	+	0.808596i	$0.700228\pi$
$462$	0	0
$463$	−28.6667	−1.33225	−0.666126	−	0.745839i	$-0.732049\pi$
$463$	−28.6667	−1.33225	−0.666126	+	0.745839i	$0.732049\pi$
$464$	0	0
$465$	−16.3208	−0.756859
$466$	0	0
$467$	1.71067	0.0791605	0.0395803	−	0.999216i	$-0.487398\pi$
$467$	1.71067	0.0791605	0.0395803	+	0.999216i	$0.487398\pi$
$468$	0	0
$469$	0.836748	0.0386374
$470$	0	0
$471$	10.1066	0.465689
$472$	0	0
$473$	16.0881	0.739730
$474$	0	0
$475$	5.03612	0.231073
$476$	0	0
$477$	13.2731	0.607736
$478$	0	0
$479$	6.60982	0.302011	0.151005	−	0.988533i	$-0.451749\pi$
$479$	6.60982	0.302011	0.151005	+	0.988533i	$0.451749\pi$
$480$	0	0
$481$	8.96012	0.408546
$482$	0	0
$483$	−0.200365	−0.00911691
$484$	0	0
$485$	−31.7704	−1.44262
$486$	0	0
$487$	33.3565	1.51153	0.755764	−	0.654844i	$-0.227266\pi$
$487$	33.3565	1.51153	0.755764	+	0.654844i	$0.227266\pi$
$488$	0	0
$489$	5.76021	0.260486
$490$	0	0
$491$	−6.25639	−0.282347	−0.141174	−	0.989985i	$-0.545088\pi$
$491$	−6.25639	−0.282347	−0.141174	+	0.989985i	$0.545088\pi$
$492$	0	0
$493$	−32.1844	−1.44951
$494$	0	0
$495$	−24.3795	−1.09578
$496$	0	0
$497$	−0.665605	−0.0298565
$498$	0	0
$499$	2.70791	0.121223	0.0606114	−	0.998161i	$-0.480695\pi$
$499$	2.70791	0.121223	0.0606114	+	0.998161i	$0.480695\pi$
$500$	0	0
$501$	−11.2839	−0.504130
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−11.2084	−0.498769
$506$	0	0
$507$	8.34994	0.370834
$508$	0	0
$509$	3.90490	0.173081	0.0865407	−	0.996248i	$-0.472419\pi$
$509$	3.90490	0.173081	0.0865407	+	0.996248i	$0.472419\pi$
$510$	0	0
$511$	−0.362571	−0.0160392
$512$	0	0
$513$	13.8119	0.609811
$514$	0	0
$515$	−36.7223	−1.61818
$516$	0	0
$517$	0.0782101	0.00343967
$518$	0	0
$519$	−10.0823	−0.442563
$520$	0	0
$521$	18.8246	0.824720	0.412360	−	0.911021i	$-0.364705\pi$
$521$	18.8246	0.824720	0.412360	+	0.911021i	$0.364705\pi$
$522$	0	0
$523$	2.22138	0.0971341	0.0485671	−	0.998820i	$-0.484535\pi$
$523$	2.22138	0.0971341	0.0485671	+	0.998820i	$0.484535\pi$
$524$	0	0
$525$	0.0789584	0.00344603
$526$	0	0
$527$	−40.0504	−1.74462
$528$	0	0
$529$	−9.01303	−0.391871
$530$	0	0
$531$	−4.91555	−0.213317
$532$	0	0
$533$	−2.73954	−0.118663
$534$	0	0
$535$	−33.8420	−1.46312
$536$	0	0
$537$	−0.497219	−0.0214566
$538$	0	0
$539$	27.3567	1.17833
$540$	0	0
$541$	−11.2074	−0.481844	−0.240922	−	0.970545i	$-0.577450\pi$
$541$	−11.2074	−0.481844	−0.240922	+	0.970545i	$0.577450\pi$
$542$	0	0
$543$	2.01410	0.0864332
$544$	0	0
$545$	12.7199	0.544859
$546$	0	0
$547$	7.53354	0.322111	0.161055	−	0.986945i	$-0.448510\pi$
$547$	7.53354	0.322111	0.161055	+	0.986945i	$0.448510\pi$
$548$	0	0
$549$	−9.61920	−0.410537
$550$	0	0
$551$	−23.7495	−1.01176
$552$	0	0
$553$	0.495132	0.0210552
$554$	0	0
$555$	−12.8124	−0.543856
$556$	0	0
$557$	−22.6294	−0.958838	−0.479419	−	0.877586i	$-0.659153\pi$
$557$	−22.6294	−0.958838	−0.479419	+	0.877586i	$0.659153\pi$
$558$	0	0
$559$	5.42854	0.229603
$560$	0	0
$561$	13.4321	0.567102
$562$	0	0
$563$	−31.3460	−1.32107	−0.660537	−	0.750793i	$-0.729671\pi$
$563$	−31.3460	−1.32107	−0.660537	+	0.750793i	$0.729671\pi$
$564$	0	0
$565$	−20.8288	−0.876275
$566$	0	0
$567$	−0.314374	−0.0132025
$568$	0	0
$569$	21.5273	0.902471	0.451235	−	0.892405i	$-0.350984\pi$
$569$	21.5273	0.902471	0.451235	+	0.892405i	$0.350984\pi$
$570$	0	0
$571$	20.9938	0.878564	0.439282	−	0.898349i	$-0.355233\pi$
$571$	20.9938	0.878564	0.439282	+	0.898349i	$0.355233\pi$
$572$	0	0
$573$	−13.1001	−0.547266
$574$	0	0
$575$	−5.51190	−0.229862
$576$	0	0
$577$	−8.66195	−0.360602	−0.180301	−	0.983612i	$-0.557707\pi$
$577$	−8.66195	−0.360602	−0.180301	+	0.983612i	$0.557707\pi$
$578$	0	0
$579$	2.58603	0.107472
$580$	0	0
$581$	−0.700930	−0.0290795
$582$	0	0
$583$	21.1888	0.877552
$584$	0	0
$585$	−8.22628	−0.340115
$586$	0	0
$587$	26.6399	1.09955	0.549773	−	0.835314i	$-0.314714\pi$
$587$	26.6399	1.09955	0.549773	+	0.835314i	$0.314714\pi$
$588$	0	0
$589$	−29.5539	−1.21775
$590$	0	0
$591$	5.53685	0.227756
$592$	0	0
$593$	−14.5238	−0.596420	−0.298210	−	0.954500i	$-0.596390\pi$
$593$	−14.5238	−0.596420	−0.298210	+	0.954500i	$0.596390\pi$
$594$	0	0
$595$	0.851106	0.0348920
$596$	0	0
$597$	−5.38473	−0.220382
$598$	0	0
$599$	−15.2683	−0.623846	−0.311923	−	0.950107i	$-0.600973\pi$
$599$	−15.2683	−0.623846	−0.311923	+	0.950107i	$0.600973\pi$
$600$	0	0
$601$	14.6966	0.599487	0.299743	−	0.954020i	$-0.403099\pi$
$601$	14.6966	0.599487	0.299743	+	0.954020i	$0.403099\pi$
$602$	0	0
$603$	28.3789	1.15568
$604$	0	0
$605$	−10.9306	−0.444392
$606$	0	0
$607$	−43.0409	−1.74698	−0.873488	−	0.486845i	$-0.838147\pi$
$607$	−43.0409	−1.74698	−0.873488	+	0.486845i	$0.838147\pi$
$608$	0	0
$609$	−0.372354	−0.0150885
$610$	0	0
$611$	0.0263901	0.00106763
$612$	0	0
$613$	−14.8163	−0.598425	−0.299212	−	0.954187i	$-0.596724\pi$
$613$	−14.8163	−0.598425	−0.299212	+	0.954187i	$0.596724\pi$
$614$	0	0
$615$	3.91737	0.157964
$616$	0	0
$617$	−13.9239	−0.560555	−0.280278	−	0.959919i	$-0.590426\pi$
$617$	−13.9239	−0.560555	−0.280278	+	0.959919i	$0.590426\pi$
$618$	0	0
$619$	15.3819	0.618250	0.309125	−	0.951021i	$-0.399964\pi$
$619$	15.3819	0.618250	0.309125	+	0.951021i	$0.399964\pi$
$620$	0	0
$621$	−15.1167	−0.606614
$622$	0	0
$623$	−0.449907	−0.0180251
$624$	0	0
$625$	−30.1969	−1.20788
$626$	0	0
$627$	9.91176	0.395838
$628$	0	0
$629$	−31.4409	−1.25363
$630$	0	0
$631$	−14.5903	−0.580831	−0.290416	−	0.956901i	$-0.593794\pi$
$631$	−14.5903	−0.580831	−0.290416	+	0.956901i	$0.593794\pi$
$632$	0	0
$633$	12.1545	0.483098
$634$	0	0
$635$	−1.20938	−0.0479928
$636$	0	0
$637$	9.23086	0.365740
$638$	0	0
$639$	−22.5745	−0.893032
$640$	0	0
$641$	30.0294	1.18609	0.593044	−	0.805170i	$-0.297926\pi$
$641$	30.0294	1.18609	0.593044	+	0.805170i	$0.297926\pi$
$642$	0	0
$643$	0.436043	0.0171959	0.00859793	−	0.999963i	$-0.497263\pi$
$643$	0.436043	0.0171959	0.00859793	+	0.999963i	$0.497263\pi$
$644$	0	0
$645$	−7.76246	−0.305646
$646$	0	0
$647$	10.5414	0.414425	0.207212	−	0.978296i	$-0.433561\pi$
$647$	10.5414	0.414425	0.207212	+	0.978296i	$0.433561\pi$
$648$	0	0
$649$	−7.84704	−0.308023
$650$	0	0
$651$	−0.463358	−0.0181604
$652$	0	0
$653$	38.6731	1.51339	0.756697	−	0.653766i	$-0.226812\pi$
$653$	38.6731	1.51339	0.756697	+	0.653766i	$0.226812\pi$
$654$	0	0
$655$	−10.2836	−0.401813
$656$	0	0
$657$	−12.2969	−0.479746
$658$	0	0
$659$	−29.8156	−1.16145	−0.580726	−	0.814099i	$-0.697231\pi$
$659$	−29.8156	−1.16145	−0.580726	+	0.814099i	$0.697231\pi$
$660$	0	0
$661$	12.4567	0.484511	0.242255	−	0.970212i	$-0.422113\pi$
$661$	12.4567	0.484511	0.242255	+	0.970212i	$0.422113\pi$
$662$	0	0
$663$	4.53234	0.176021
$664$	0	0
$665$	0.628046	0.0243546
$666$	0	0
$667$	25.9931	1.00646
$668$	0	0
$669$	−14.6366	−0.565882
$670$	0	0
$671$	−15.3558	−0.592804
$672$	0	0
$673$	13.8743	0.534814	0.267407	−	0.963584i	$-0.413833\pi$
$673$	13.8743	0.534814	0.267407	+	0.963584i	$0.413833\pi$
$674$	0	0
$675$	5.95711	0.229289
$676$	0	0
$677$	20.1594	0.774788	0.387394	−	0.921914i	$-0.373375\pi$
$677$	20.1594	0.774788	0.387394	+	0.921914i	$0.373375\pi$
$678$	0	0
$679$	−0.901982	−0.0346149
$680$	0	0
$681$	−10.8967	−0.417561
$682$	0	0
$683$	−19.5990	−0.749934	−0.374967	−	0.927038i	$-0.622346\pi$
$683$	−19.5990	−0.749934	−0.374967	+	0.927038i	$0.622346\pi$
$684$	0	0
$685$	19.6169	0.749523
$686$	0	0
$687$	15.1073	0.576379
$688$	0	0
$689$	7.14968	0.272381
$690$	0	0
$691$	−7.48363	−0.284691	−0.142345	−	0.989817i	$-0.545464\pi$
$691$	−7.48363	−0.284691	−0.142345	+	0.989817i	$0.545464\pi$
$692$	0	0
$693$	−0.692148	−0.0262925
$694$	0	0
$695$	−24.5576	−0.931521
$696$	0	0
$697$	9.61302	0.364119
$698$	0	0
$699$	−5.06490	−0.191572
$700$	0	0
$701$	22.4673	0.848580	0.424290	−	0.905526i	$-0.360524\pi$
$701$	22.4673	0.848580	0.424290	+	0.905526i	$0.360524\pi$
$702$	0	0
$703$	−23.2008	−0.875035
$704$	0	0
$705$	−0.0377362	−0.00142123
$706$	0	0
$707$	−0.318215	−0.0119677
$708$	0	0
$709$	19.9426	0.748960	0.374480	−	0.927235i	$-0.377821\pi$
$709$	19.9426	0.748960	0.374480	+	0.927235i	$0.377821\pi$
$710$	0	0
$711$	16.7928	0.629778
$712$	0	0
$713$	32.3459	1.21136
$714$	0	0
$715$	−13.1322	−0.491116
$716$	0	0
$717$	8.72010	0.325658
$718$	0	0
$719$	25.7255	0.959398	0.479699	−	0.877433i	$-0.340746\pi$
$719$	25.7255	0.959398	0.479699	+	0.877433i	$0.340746\pi$
$720$	0	0
$721$	−1.04257	−0.0388274
$722$	0	0
$723$	16.3286	0.607265
$724$	0	0
$725$	−10.2432	−0.380423
$726$	0	0
$727$	−1.02175	−0.0378945	−0.0189473	−	0.999820i	$-0.506031\pi$
$727$	−1.02175	−0.0378945	−0.0189473	+	0.999820i	$0.506031\pi$
$728$	0	0
$729$	−1.66886	−0.0618096
$730$	0	0
$731$	−19.0487	−0.704540
$732$	0	0
$733$	−33.7807	−1.24772	−0.623859	−	0.781537i	$-0.714436\pi$
$733$	−33.7807	−1.24772	−0.623859	+	0.781537i	$0.714436\pi$
$734$	0	0
$735$	−13.1995	−0.486872
$736$	0	0
$737$	45.3032	1.66877
$738$	0	0
$739$	21.3291	0.784605	0.392303	−	0.919836i	$-0.371679\pi$
$739$	21.3291	0.784605	0.392303	+	0.919836i	$0.371679\pi$
$740$	0	0
$741$	3.34449	0.122863
$742$	0	0
$743$	−22.9522	−0.842034	−0.421017	−	0.907053i	$-0.638327\pi$
$743$	−22.9522	−0.842034	−0.421017	+	0.907053i	$0.638327\pi$
$744$	0	0
$745$	8.52702	0.312406
$746$	0	0
$747$	−23.7725	−0.869791
$748$	0	0
$749$	−0.960796	−0.0351067
$750$	0	0
$751$	36.7362	1.34052	0.670262	−	0.742124i	$-0.266182\pi$
$751$	36.7362	1.34052	0.670262	+	0.742124i	$0.266182\pi$
$752$	0	0
$753$	13.6428	0.497170
$754$	0	0
$755$	0.187377	0.00681935
$756$	0	0
$757$	8.80021	0.319849	0.159925	−	0.987129i	$-0.448875\pi$
$757$	8.80021	0.319849	0.159925	+	0.987129i	$0.448875\pi$
$758$	0	0
$759$	−10.8481	−0.393763
$760$	0	0
$761$	20.7859	0.753488	0.376744	−	0.926317i	$-0.377044\pi$
$761$	20.7859	0.753488	0.376744	+	0.926317i	$0.377044\pi$
$762$	0	0
$763$	0.361125	0.0130736
$764$	0	0
$765$	28.8659	1.04365
$766$	0	0
$767$	−2.64780	−0.0956065
$768$	0	0
$769$	21.7495	0.784307	0.392154	−	0.919900i	$-0.371730\pi$
$769$	21.7495	0.784307	0.392154	+	0.919900i	$0.371730\pi$
$770$	0	0
$771$	7.96775	0.286951
$772$	0	0
$773$	−26.0955	−0.938591	−0.469295	−	0.883041i	$-0.655492\pi$
$773$	−26.0955	−0.938591	−0.469295	+	0.883041i	$0.655492\pi$
$774$	0	0
$775$	−12.7467	−0.457874
$776$	0	0
$777$	−0.363752	−0.0130495
$778$	0	0
$779$	7.09361	0.254155
$780$	0	0
$781$	−36.0372	−1.28951
$782$	0	0
$783$	−28.0927	−1.00395
$784$	0	0
$785$	34.6722	1.23750
$786$	0	0
$787$	27.0446	0.964035	0.482018	−	0.876161i	$-0.339904\pi$
$787$	27.0446	0.964035	0.482018	+	0.876161i	$0.339904\pi$
$788$	0	0
$789$	−16.7899	−0.597736
$790$	0	0
$791$	−0.591343	−0.0210258
$792$	0	0
$793$	−5.18145	−0.183999
$794$	0	0
$795$	−10.2236	−0.362593
$796$	0	0
$797$	34.9781	1.23899	0.619495	−	0.785001i	$-0.287338\pi$
$797$	34.9781	1.23899	0.619495	+	0.785001i	$0.287338\pi$
$798$	0	0
$799$	−0.0926026	−0.00327604
$800$	0	0
$801$	−15.2589	−0.539147
$802$	0	0
$803$	−19.6303	−0.692739
$804$	0	0
$805$	−0.687379	−0.0242269
$806$	0	0
$807$	19.6539	0.691852
$808$	0	0
$809$	−6.13748	−0.215782	−0.107891	−	0.994163i	$-0.534410\pi$
$809$	−6.13748	−0.215782	−0.107891	+	0.994163i	$0.534410\pi$
$810$	0	0
$811$	−16.1036	−0.565475	−0.282738	−	0.959197i	$-0.591243\pi$
$811$	−16.1036	−0.565475	−0.282738	+	0.959197i	$0.591243\pi$
$812$	0	0
$813$	−18.9727	−0.665400
$814$	0	0
$815$	19.7612	0.692205
$816$	0	0
$817$	−14.0563	−0.491769
$818$	0	0
$819$	−0.233549	−0.00816087
$820$	0	0
$821$	−8.25056	−0.287947	−0.143973	−	0.989582i	$-0.545988\pi$
$821$	−8.25056	−0.287947	−0.143973	+	0.989582i	$0.545988\pi$
$822$	0	0
$823$	−22.3216	−0.778080	−0.389040	−	0.921221i	$-0.627193\pi$
$823$	−22.3216	−0.778080	−0.389040	+	0.921221i	$0.627193\pi$
$824$	0	0
$825$	4.27497	0.148835
$826$	0	0
$827$	−33.4012	−1.16147	−0.580736	−	0.814092i	$-0.697235\pi$
$827$	−33.4012	−1.16147	−0.580736	+	0.814092i	$0.697235\pi$
$828$	0	0
$829$	36.0324	1.25146	0.625728	−	0.780041i	$-0.284802\pi$
$829$	36.0324	1.25146	0.625728	+	0.780041i	$0.284802\pi$
$830$	0	0
$831$	−17.1503	−0.594936
$832$	0	0
$833$	−32.3909	−1.12228
$834$	0	0
$835$	−38.7112	−1.33965
$836$	0	0
$837$	−34.9586	−1.20835
$838$	0	0
$839$	39.6958	1.37045	0.685226	−	0.728330i	$-0.259703\pi$
$839$	39.6958	1.37045	0.685226	+	0.728330i	$0.259703\pi$
$840$	0	0
$841$	19.3051	0.665693
$842$	0	0
$843$	−17.0807	−0.588290
$844$	0	0
$845$	28.6456	0.985439
$846$	0	0
$847$	−0.310326	−0.0106629
$848$	0	0
$849$	−14.9175	−0.511967
$850$	0	0
$851$	25.3926	0.870448
$852$	0	0
$853$	−0.569322	−0.0194932	−0.00974661	−	0.999953i	$-0.503102\pi$
$853$	−0.569322	−0.0194932	−0.00974661	+	0.999953i	$0.503102\pi$
$854$	0	0
$855$	21.3006	0.728466
$856$	0	0
$857$	−14.5414	−0.496723	−0.248362	−	0.968667i	$-0.579892\pi$
$857$	−14.5414	−0.496723	−0.248362	+	0.968667i	$0.579892\pi$
$858$	0	0
$859$	−9.03745	−0.308354	−0.154177	−	0.988043i	$-0.549273\pi$
$859$	−9.03745	−0.308354	−0.154177	+	0.988043i	$0.549273\pi$
$860$	0	0
$861$	0.111217	0.00379025
$862$	0	0
$863$	−0.814226	−0.0277166	−0.0138583	−	0.999904i	$-0.504411\pi$
$863$	−0.814226	−0.0277166	−0.0138583	+	0.999904i	$0.504411\pi$
$864$	0	0
$865$	−34.5887	−1.17605
$866$	0	0
$867$	−3.29568	−0.111927
$868$	0	0
$869$	26.8074	0.909380
$870$	0	0
$871$	15.2865	0.517963
$872$	0	0
$873$	−30.5914	−1.03536
$874$	0	0
$875$	−0.648099	−0.0219097
$876$	0	0
$877$	−43.5350	−1.47007	−0.735036	−	0.678028i	$-0.762835\pi$
$877$	−43.5350	−1.47007	−0.735036	+	0.678028i	$0.762835\pi$
$878$	0	0
$879$	17.3044	0.583664
$880$	0	0
$881$	10.2033	0.343757	0.171878	−	0.985118i	$-0.445016\pi$
$881$	10.2033	0.343757	0.171878	+	0.985118i	$0.445016\pi$
$882$	0	0
$883$	3.07095	0.103346	0.0516728	−	0.998664i	$-0.483545\pi$
$883$	3.07095	0.103346	0.0516728	+	0.998664i	$0.483545\pi$
$884$	0	0
$885$	3.78618	0.127271
$886$	0	0
$887$	−39.5415	−1.32767	−0.663836	−	0.747878i	$-0.731073\pi$
$887$	−39.5415	−1.32767	−0.663836	+	0.747878i	$0.731073\pi$
$888$	0	0
$889$	−0.0343351	−0.00115156
$890$	0	0
$891$	−17.0208	−0.570220
$892$	0	0
$893$	−0.0683331	−0.00228668
$894$	0	0
$895$	−1.70578	−0.0570179
$896$	0	0
$897$	−3.66045	−0.122219
$898$	0	0
$899$	60.1110	2.00481
$900$	0	0
$901$	−25.0881	−0.835806
$902$	0	0
$903$	−0.220381	−0.00733382
$904$	0	0
$905$	6.90964	0.229684
$906$	0	0
$907$	18.9114	0.627943	0.313971	−	0.949432i	$-0.398340\pi$
$907$	18.9114	0.627943	0.313971	+	0.949432i	$0.398340\pi$
$908$	0	0
$909$	−10.7925	−0.357964
$910$	0	0
$911$	−1.23286	−0.0408463	−0.0204232	−	0.999791i	$-0.506501\pi$
$911$	−1.23286	−0.0408463	−0.0204232	+	0.999791i	$0.506501\pi$
$912$	0	0
$913$	−37.9497	−1.25595
$914$	0	0
$915$	7.40914	0.244939
$916$	0	0
$917$	−0.291957	−0.00964128
$918$	0	0
$919$	27.7336	0.914848	0.457424	−	0.889249i	$-0.348772\pi$
$919$	27.7336	0.914848	0.457424	+	0.889249i	$0.348772\pi$
$920$	0	0
$921$	−18.1442	−0.597870
$922$	0	0
$923$	−12.1599	−0.400248
$924$	0	0
$925$	−10.0066	−0.329014
$926$	0	0
$927$	−35.3595	−1.16136
$928$	0	0
$929$	37.6994	1.23688	0.618438	−	0.785833i	$-0.287766\pi$
$929$	37.6994	1.23688	0.618438	+	0.785833i	$0.287766\pi$
$930$	0	0
$931$	−23.9018	−0.783351
$932$	0	0
$933$	−20.5200	−0.671796
$934$	0	0
$935$	46.0806	1.50700
$936$	0	0
$937$	18.7918	0.613901	0.306951	−	0.951725i	$-0.400691\pi$
$937$	18.7918	0.613901	0.306951	+	0.951725i	$0.400691\pi$
$938$	0	0
$939$	−10.8429	−0.353845
$940$	0	0
$941$	44.0432	1.43577	0.717883	−	0.696164i	$-0.245111\pi$
$941$	44.0432	1.43577	0.717883	+	0.696164i	$0.245111\pi$
$942$	0	0
$943$	−7.76376	−0.252823
$944$	0	0
$945$	0.742901	0.0241666
$946$	0	0
$947$	−10.4978	−0.341132	−0.170566	−	0.985346i	$-0.554560\pi$
$947$	−10.4978	−0.341132	−0.170566	+	0.985346i	$0.554560\pi$
$948$	0	0
$949$	−6.62379	−0.215017
$950$	0	0
$951$	19.0036	0.616233
$952$	0	0
$953$	−43.6480	−1.41390	−0.706949	−	0.707264i	$-0.749929\pi$
$953$	−43.6480	−1.41390	−0.706949	+	0.707264i	$0.749929\pi$
$954$	0	0
$955$	−44.9418	−1.45428
$956$	0	0
$957$	−20.1600	−0.651679
$958$	0	0
$959$	0.556936	0.0179844
$960$	0	0
$961$	43.8023	1.41298
$962$	0	0
$963$	−32.5861	−1.05007
$964$	0	0
$965$	8.87173	0.285591
$966$	0	0
$967$	29.4400	0.946728	0.473364	−	0.880867i	$-0.343040\pi$
$967$	29.4400	0.946728	0.473364	+	0.880867i	$0.343040\pi$
$968$	0	0
$969$	−11.7358	−0.377007
$970$	0	0
$971$	7.98084	0.256117	0.128059	−	0.991767i	$-0.459125\pi$
$971$	7.98084	0.256117	0.128059	+	0.991767i	$0.459125\pi$
$972$	0	0
$973$	−0.697204	−0.0223513
$974$	0	0
$975$	1.44249	0.0461966
$976$	0	0
$977$	−9.84120	−0.314848	−0.157424	−	0.987531i	$-0.550319\pi$
$977$	−9.84120	−0.314848	−0.157424	+	0.987531i	$0.550319\pi$
$978$	0	0
$979$	−24.3588	−0.778512
$980$	0	0
$981$	12.2478	0.391042
$982$	0	0
$983$	−28.7754	−0.917792	−0.458896	−	0.888490i	$-0.651755\pi$
$983$	−28.7754	−0.917792	−0.458896	+	0.888490i	$0.651755\pi$
$984$	0	0
$985$	18.9949	0.605229
$986$	0	0
$987$	−0.00107135	−3.41016e−5 0
$988$	0	0
$989$	15.3843	0.489191
$990$	0	0
$991$	−42.1118	−1.33773	−0.668863	−	0.743386i	$-0.733219\pi$
$991$	−42.1118	−1.33773	−0.668863	+	0.743386i	$0.733219\pi$
$992$	0	0
$993$	−8.06855	−0.256048
$994$	0	0
$995$	−18.4731	−0.585635
$996$	0	0
$997$	−13.7840	−0.436544	−0.218272	−	0.975888i	$-0.570042\pi$
$997$	−13.7840	−0.436544	−0.218272	+	0.975888i	$0.570042\pi$
$998$	0	0
$999$	−27.4437	−0.868279

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.g.1.12	✓	33
4.3	odd	2		8048.2.a.x.1.22		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.12	✓	33	1.1	even	1	trivial
8048.2.a.x.1.22		33	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-0.741660 q^{3} -2.54437 q^{5} -0.0722362 q^{7} -2.44994 q^{9} +O(q^{10})\)	\(q-0.741660 q^{3} -2.54437 q^{5} -0.0722362 q^{7} -2.44994 q^{9} -3.91101 q^{11} -1.31968 q^{13} +1.88705 q^{15} +4.63073 q^{17} +3.41710 q^{19} +0.0535746 q^{21} -3.73992 q^{23} +1.47380 q^{25} +4.04200 q^{27} -6.95019 q^{29} -8.64883 q^{31} +2.90064 q^{33} +0.183795 q^{35} -6.78962 q^{37} +0.978752 q^{39} +2.07592 q^{41} -4.11353 q^{43} +6.23355 q^{45} -0.0199974 q^{47} -6.99478 q^{49} -3.43442 q^{51} -5.41774 q^{53} +9.95104 q^{55} -2.53432 q^{57} +2.00640 q^{59} +3.92630 q^{61} +0.176974 q^{63} +3.35775 q^{65} -11.5835 q^{67} +2.77374 q^{69} +9.21429 q^{71} +5.01924 q^{73} -1.09306 q^{75} +0.282516 q^{77} -6.85436 q^{79} +4.35203 q^{81} +9.70331 q^{83} -11.7823 q^{85} +5.15467 q^{87} +6.22827 q^{89} +0.0953285 q^{91} +6.41449 q^{93} -8.69435 q^{95} +12.4866 q^{97} +9.58174 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * q^99

Introduction

L-functions

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